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Wave Motion

Wave Motion. Energy and power in sinusoidal waves. Energy in Waves. as waves propagate through a medium, they transport energy eg: ship moving up and down on a lake eg: feeling sound waves at a rock concert hence, we can talk about energy and the ‘rate of energy transfer’. Energy and Power.

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Wave Motion

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  1. Wave Motion • Energy and power in sinusoidal waves Physics 1B03summer-Lecture 9

  2. Energy in Waves • as waves propagate through a medium, they transport energyeg: ship moving up and down on a lakeeg: feeling sound waves at a rock concert • hence, we can talk about energy and the ‘rate of energy transfer’ Physics 1B03summer-Lecture 9

  3. Energy and Power A stretched rope has energy/unit length: ds dm dx For small A and large l, we can ignore the difference between “ds”, “dx” : dm = μ dx (μ= mass/unit length) Physics 1B03summer-Lecture 9

  4. The mass dm vibrates in simple harmonic motion. Its maximum kinetic energy is dKmax = ½(dm)vmax2 = ½(dm)(ωA)2 The average kinetic energy is half this maximum value, but there is also an equal amount of potential energy in the wave. The total energy (kinetic plus potential) is therefore: dE = ½(dm) ω 2A2 To get the energy per unit length (or energy ‘density’), replace the mass dm with the mass per unit length m: Physics 1B03summer-Lecture 9

  5. Power: Energy travels at the wave speed v, So waves on a string, Both the energy density and the power transmitted are proportional to the square of the amplitude. This is a general property of sinusoidal waves. Physics 1B03summer-Lecture 9

  6. Example A string for which μ=5.0x10-2 kg/m is under tension of 80.0 N. How much power must be supplied to the string to generate sinusoidal waves at a frequency of 60Hz and with an amplitude of 6.0 cm ? Physics 1B03summer-Lecture 9

  7. Example A sinusoidal wave on a string is described by the equation: y(x,t) = (0.15m)sin(0.80x-50t) where x is in meters and t in seconds. If μ=12.0g/m, determine: • the speed of the wave • the speed of particles on the wave at any time • the wavelength • the frequency • the power transmitted to the wave Physics 1B03summer-Lecture 9

  8. Concept Quiz The sound waves from your 100-watt stereo causes windows across the street to vibrate with an amplitude of 1 mm. If you use a 400-watt amplifier, what sort of amplitude can you get from the windows? • 2mm • 4mm • 16 mm Physics 1B03summer-Lecture 9

  9. detectors (areaA) source Intensity I= Power per unit area Unit: W / m2 • (the area is measured perpendicular to the wave velocity) Intensity ~ (amplitude)2 Physics 1B03summer-Lecture 9

  10. How would the intensity depend on distance from the source for: • waves spreading out equally in all directions in space? (This is called an“isotropic” source, or a source of “spherical waves”.) • Waves spreading out on a two-dimensional surface, e.g., circular ripples from a stone dropped into water? How would the amplitude depend on distance? Physics 1B03summer-Lecture 9

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